Relative Bias Under Imperfect Identification in Observational Causal Inference

Abstract

To conduct causal inference in observational settings, researchers must rely on certain identifying assumptions. In practice, these assumptions are unlikely to hold exactly. This paper considers the bias of selection-on-observables, instrumental variables, and proximal inference estimates under violations of their identifying assumptions. We develop bias expressions for IV and proximal inference that show how violations of their respective assumptions are amplified by any unmeasured confounding in the outcome variable. We propose a set of sensitivity tools that quantify the sensitivity of different identification strategies, and an augmented bias contour plot visualizes the relationship between these strategies. We argue that the act of choosing an identification strategy implicitly expresses a belief about the degree of violations that must be present in alternative identification strategies. Even when researchers intend to conduct an IV or proximal analysis, a sensitivity analysis comparing different identification strategies can help to better understand the implications of each set of assumptions. Throughout, we compare the different approaches on a re-analysis of the impact of state surveillance on the incidence of protest in Communist Poland.

Figures (6)

• Figure 1: Selection-on-observables
• Figure 2: Instrumental variables
• Figure 3: Proximal inference
• Figure 5: Robustness value by identification strategy for running example as $\rho_3 = R_{Z\sim U\mid W_Z, W_Y}$ is constrained to different values along its possible range. The overall robustness value is the minimum along each curve. The parameter $\rho_4=R_{Y\sim U\mid Z, W_Z, W_Y}$ decreases monotonically from 1 to 0 as $\rho_3$ increases along the x-axis.
• Figure 6: Sensitivity plot for the running example. The contour lines are on a logarithmic scale and show the amount of SOO confounding bias. Contours are colored according to the regions where each estimator has lowest bias. Gray crosses indicate benchmark sensitivity parameter values for each of the covariates.

Theorems & Definitions (19)

• Proposition 2.1: SOO validity
• Proposition 2.2: IV validity
• Theorem 3.1: Bias of IV estimate
• Theorem 3.2: Bias of the proximal estimator
• Proposition A.1: Proximal validity
• Corollary A.1: Relative bias with exogenous instruments
• Corollary A.2
• Corollary A.3
• Corollary A.4
• Lemma D.1